## Thursday, March 10, 2011

### Summary of Potential energy...

If a particle of mass m is at a distance y above the Earth’s surface, the gravitational potential
energy of the particle–Earth system is

(8.2)
The elastic potential energy stored in a spring of force constant k is

(8.11)
A reference configuration of the system should be chosen, and this configuration is often
assigned a potential energy of zero.

A force is conservative if the work it does on a particle moving between two points
is independent of the path the particle takes between the two points. Furthermore, a
force is conservative if the work it does on a particle is zero when the particle moves
through an arbitrary closed path and returns to its initial position. A force that does
not meet these criteria is said to be nonconservative.

The total mechanical energy of a system is defined as the sum of the kinetic energy
and the potential energy:

If a system is isolated and if no nonconservative forces are acting on objects inside the
system, then the total mechanical energy of the system is constant:
(8.9)

If nonconservative forces (such as friction) act on objects inside a system, then mechanical
energy is not conserved. In these situations, the difference between the total
final mechanical energy and the total initial mechanical energy of the system equals
the energy transformed to internal energy by the nonconservative forces.

A potential energy function U can be associated only with a conservative force. If
a conservative force F acts between members of a system while one member moves
along the x axis from xi to xf , then the change in the potential energy of the system
equals the negative of the work done by that force:

(8.16)

Systems can be in three types of equilibrium configurations when the net force on a
member of the system is zero. Configurations of stable equilibrium correspond to
those for which U(x) is a minimum. Configurations of unstable equilibrium correspond
to those for which U(x) is a maximum. Neutral equilibrium arises where U is
constant as a member of the system moves over some region.